The reduced value of the fraction $\frac{m}{n}$ is $\frac{1}{4}$. The reduced value of the fraction $\frac{m^2}{n}$ is 2. What is the value of $m+n$?
Explanation: Observe that \[ \frac{m}{n}\cdot m = \frac{m^2}{n}. \] Since $m/n=1/4$ and $m^2/n=2$, $m$ must be $8$. Substituting for $m$ in $m/n=1/4$, we find $n=32$. Therefore, $m+n=8+32=\boxed{40}$.